Inverse problems of the Erdős-Ko-Rado type theorems for families of vector spaces and permutations

Abstract

Ever since the famous Erdős-Ko-Rado theorem initiated the study of intersecting families of subsets, extremal problems regarding intersecting properties of families of various combinatorial objects have been extensively investigated. Among them, studies about families of subsets, vector spaces and permutations are of particular concerns. Recently, we proposed a new quantitative intersection problem for families of subsets: For ${\cal F} \subseteq \left({\matrix{{[n]} \cr k \cr}} \right)$, define its total intersection number as ${\cal I}({\cal F}) = \sum\nolimits_{{F_1},{F_2} \in {\cal F}} {\left| {{F_1} \cap {F_2}} \right|}$. Then, what is the structure of ${\cal F}$ when it has the maximal total intersection number among all the families in $\left({\matrix{{[n]} \cr k \cr}} \right)$ with the same family size? In a recent paper, Kong and Ge (2020) studied this problem and characterized extremal structures of families maximizing the total intersection number of given sizes. In this paper, we consider the analogues of this problem for families of vector spaces and permutations. For certain ranges of family size, we provide structural characterizations for both families of subspaces and families of permutations having maximal total intersection numbers. To some extent, these results determine the unique structure of the optimal family for some certain values of $\left| {\cal F} \right|$ and characterize the relationship between having maximal total intersection number and being intersecting. Besides, we also show several upper bounds on the total intersection numbers for both families of subspaces and families of permutations of given sizes.